2001/04/01 by Tom Leinster, Leinster, Tom · 1 citation
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT
paper · pdf · doi:10.48550/arxiv.math/0104012
12 pages. Written to be comprehensible to an undergraduate readership
arxiv created 2001/04/01 · openalex publication_date 2001/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not arise.) The notion of perfect group generalizes that of perfect number, since a cyclic group is perfect exactly when its order is perfect. We show that, in fact, the only abelian perfect groups are the cyclic ones, and exhibit some non-abelian perfect groups of even order.